Pgm: A Python package for free energy calculations within the phonon gas model

Abstract

The quasi-harmonic approximation (QHA) is a powerful method that uses the volume dependence of non-interacting phonons to compute the free energy of materials at high pressures (P) and temperatures (T). However, anharmonicity, electronic excitations in metals, or both, introduce an intrinsic T-dependence on phonon frequencies, rendering the QHA inadequate. Here we present a Python code, pgm, to compute the free energy and thermodynamic property within the phonon gas model (PGM) that uses T-dependent phonon quasiparticle frequencies. In this case, the vibrational contribution to the Helmholtz free energy is obtained by integrating the vibrational entropy, which can be readily calculated for a system of phonon quasiparticles. Other thermodynamic properties are then obtained from standard thermodynamic relations. We demonstrate the successful applications of pgm to two cases of geophysical significance: cubic CaSiO3-perovskite (cCaPv), a strongly anharmonic insulator and the third most abundant phase of the Earth's lower mantle, and NiAs-type (B8) FeO, a partially covalent-metallic system. This is the oxide endmember of a recently discovered iron-rich FenO alloy phase likely to exist in the Earth's inner core. Program summaryProgram Title:pgmCPC Library link to program files:https://doi.org/10.17632/8rfw6syvzp.1Developer's repository link:https://github.com/MineralsCloud/pgmLicensing provisions: GNU General Public License 3Programming language: Python3Nature of problem: The classic quasi-harmonic approximation (QHA) method to compute the vibrational free energy does not apply to physical systems when phonon frequencies have an intrinsic and non-negligible temperature (T) dependence. Examples are anharmonic systems or metals with abundant electronic thermal excitations. Both cases introduce an intrinsic T-dependence on phonon frequencies.Solution method: The method implemented in pgm is based on the phonon gas model where the entropy is well defined for T-dependent phonon quasiparticle. The free energy is calculated by integrating the entropy, making it suitable for anharmonic systems or systems with extensive thermal electronic excitations affecting phonon frequencies. The static free energy, the vibrational density of states (VDoS), and the entropy are first computed on sparse T- and P-grids. The entropy is then suitably interpolated on denser user-specified grids for integration.Additional comments, including restrictions and unusual features: The package allows it to be run directly in the command line. It can also be incorporated into other programs. We implemented Just-in-time (JIT) compiling and parallel computing techniques [1] in pgm Python code to speed up the numerical calculation.